Is there a "right" way to weigh available stats for fantasy games?

There is an abundance of systems used to evaluate stats for fantasy games around the web. With 1 Point = 1 Fantasy Point, I've seen the following systems being used:

P R A B S TO 3ma FGmi FT mi Sh. Bonus

Sandbox 98-99 1 2 2 2 2 -1 1 0 0 Yes

Sandbox 97-98 1 2 2 2 2 -2 1 0 0 Yes

Hoopscity (Fantasy) 1 1 1 1 1 -2 1 0 0 No

Hoopscity (H-to-H) 1 1 1 1 1 -1 0 0 0 Yes

Small World/CNN? 1 1.5 2 2.5 2.5 -1 0 -0.5 -1 No

NBA.com/ESPN 1 1 1 1 1 -1 0 -1 -1 No

This analysis 1 1 * 1+ 1.5+ -1.5- 0.75 -1 -0.5 No

So, is there a "right" way to weigh available stats for fantasy games? Or at least, can we put some bounds on the acceptable values of the weights? Or is it ok for everything to be completely arbitrary, and we should accept systems like Hoopscity's Fantasy where a TO counted for -2 but a steal for +1, or -1 and +2, respectively, in Sandbox 98-99, and -1 and +2.5, respectively, in SmallWorld!?

An interesting example comes from the case of a player who is alone under the basket, but misses the layup. Gets the rebound and misses again, and again, and then he is blocked in the 4th attempt. Under the Sandbox system this player would get 6 FPs, for the 3 offensive rebounds (1 of which could be uncontested), with an adverse effect in the small FG% bonus (practically max 3) he could get. In my book, this player should NOT get more points than had he made the basket the very first time (2 FPs under the Sandbox system)! So we need a CONSISTENT system, and Sandbox and Hoopscity fall short in this category by default.

The same issues of fairness and consistency pertain to evaluating players for purposes other than fantasy games, although the limitations of currently kept stats are obvious since they do not include some measurable stats, like forced TOs, fouls against, participation in team rebounds and steals, points scored by opponent guarded etc (so there is much ground for improvement), as well as some that can not be measured, like helping on defense, placing picks on ofense, making plays and FTs when it counts etc.

Shooting % bonuses (as a substitute for made or missed FGs and FTs) are obviously absolutely arbitrary and therefore wrong. They are just used to make systems "fancy", and besides being arbtrary they are sometimes grossly abused (as by Sandbox that gives some bonus for 5/11 but not for 5/5 shooting).

The way to approach this problem is to relate each statistical category to a common "quantity", and a couple of convenient ones are "ball possession" and "point scored".

1) MISSED FGs and OFFENSIVE REBOUNDS

The observed ratio of defensive to offensive rebounds is approximately 2:1. Hence, ignoring the balls that end out-of-bounds without being rebounded, possession after about 1 out of 3 missed FGs is renewed by an offensive rebound. Hence

offensive rebound = |missed FG| = 2/3 possession

Note: the "absolute value" is used because a missed FG has a negative value.

Although, in general, an offensive rebound is about twice more difficult to win and it is considered more important than a defensive one, in terms of possessions they are equally important. Also, offensive rebounds after blocks or airballs give possessions with limited time left, so there are factors acting in opposite directions. Hence, given that they are often lumped together in stat sheets, we assume that

rebound = defensive rebound = offensive rebound

TOs may occur from offensive fouls, or passes out-of-bounds, which generally mean just a loss of possession. On the other hand, they may result from steals that may lead to a fast-break, but this does not show on the stats sheet. On the other hand some TOs may not really be the player's fault, as in the case of good passes (by all reasonable standards) that are not handled by players who should normally do so. So again there are factors acting in opposite dirctions, but in generally a TO should have at least the value of a possession. Steals should have at least the value of a TO, since the ball always is in play (not out-of-bounds), so the chances for a fast-break are higher. Hence

steal = |TO| + x = possession + x + y

A made FG constitutes loss of possession, hence

|made FG| = possession

A missed or made FT in the first attempt (when shooting two FTs) is immaterial as far as possession of the ball is concerned. A missed or made FT in the last attempt is treated as a missed of made FG. It is assumed that half the missed or made FTs are on the last attempt and half on the first. The effects of single and triple FTs are ignored since they are relatively few and impossible to get from usual stat sheets. Hence

|made FT| = 1/2 |made FG|

|missed FT| = 1/2 |missed FG|

The average point output is about 0.85 (~ 6/7) points scored per ball possession (taking into account turnovers).

6/7 points = 1 possession

Note that up to this point MADE FGs and FTs have NEGATIVE value because they constitute loss of possession. We will later lump this value with that of points scored.

An assist assures that the output of a possession is at least 2 points instead of the average 6/7. Hence its additive value should be 2 - 6/7 = 8/7 points for 2p FGs and 3 - 6/7 = 15/7 points for 3p FGs. Assuming that the ratio of assisted 3p FGs to assisted FGs is the same with the ratio of total made 3p FGs to total made FGs, let's say 15% (ranges between about 10% -e.g. Spurs- and 20% -e.g. Rockets-), the average value of an assist is about 9/7 (8 x 0.85 + 15 x 0.15 = 9.05) of a point. The question is, should all this additive value be awarded to the passer? I'm not positive about that. Hence

assist = 9/7 points - z

Since at least half of the additive value of the assist should be credited to the passer, z is bound between 0 and 1/2 assists.

I think that blocks are the most difficult stat to assign a value to. To begin with, we have the intimidation factor that forces offenses away of ordinary plays. But setting that aside, for the moment, let's examine the value of a block made not by a career blocker (e.g. Mutombo), but by any player (e.g. Kerr). We can assume that by those blocks that do not end out-of-bounds, defense wins about 2/3 (similar to rebounds), However often the blocked player has an advantage. Hence, the net value of the block would be

block = 2/3 possession - w1

Including the intimidation factor and the "changed shots" that are often caused by blockers, we end up with a rough estimate of

block = 2/3 possession - w1 + w2 = 2/3 possession + w

w is highly arbitrary

Assigning a base value of

1 point = 7 fantasy points (FP)

in order to have all stats in integer form, we get:

9-z assist (0< z < 4.5)

10) LUMPING PENALTYS FOR MADE FGs AND FTs WITH POINTS SCORED

The negative value of MADE FGs and FTs (cince they result in lost ball possessions) is balanced by the high value of points. If we lump all of them together to avoid the unconventional subtraction of points for made FGs and FTs, we have:

2p shots = 2 x 7 - 6 = 8 FP, or 1 point = 4 FP (same for FTs under the assumptions mentioned in Section 5)

Using the value of 1 point = 4 FP, we examine 3-p shots:

3p shots = 3 x 7 - 6 = 15 FP = 3 x 4 + [extra value of 3-p shot = 3]

This shows that there is a THEORETICALLY derived extra value to made 3-pointers when made FGs are not accounted in the stats! :-)

Hence, the above system can be rewritten as

9-z assist (0 < z < 4.5)

6+x+y steal

4+w block

4 rebound

4 point

3 made 3

-2 missed FT

-4 missed FG

-6-y TO

11) SCALING

If we divide everything by 4 to get to the usual 1 point = 1 FP, we get (using the same letter symbols for convenience):

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

@ 2.25-z assist (0 < z < 2.25/2) @

@ 1.50+x+y steal @

@ 1.00+w block @

@ 1 rebound @

@ 1 point @

@ 0.75 made 3 @

@ -0.50 missed FT @

@ -1.00 missed FG @

@ -1.50-y TO @

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

12) EXAMPLE

The values of x, y, z, and w are SOMEWHAT arbitrary. I'm sure that a more thorough analysis could derive more narrow bounds for them, as well as an upper bound for x+y (and the value of a steal). Some good choices, in my opinion, are:

x = 0.5 (steals always occur in bounds and lead to fast-breaks more often that TOs)

y = 0.5 (some TOs lead to fast-breaks, others are offensive fouls etc)

z = 0.25 (this arbitrarily assigns 8/9 of the additive value of an assist to the passer, and the rest 1/9 -unclaimed- to the recipient of the pass)

w = 1.5 (highly arbitrary)

resulting in

2.5 block (1 =< block)

2.5 steal (1.5 =< |TO| =< steal)

2 assist (1.125 =< assist =< 2.5)

1 rebound

1 point

0.75 3 made

-0.5 FT missed

-1 FG missed

-2 TO (1.5 =< |TO| =< steal)

13) COMPARISON WITH SYSTEMS USED

It compares quite well with most of the systems, which is a strong indication that the above "theoretical" analysis is close to the practical considerations probably used to develop these systems.

The closest system is the one used by Small World. However it fall shorts in:

1) Not assigning a value to 3s made

2) Reversing the penalties for missed FTs (-1) and FGs (-0.5)

3) Setting the penalty for TOs (-1) a too low

4) Setting the value for a rebound (2) too high

14) GRADING THE STAT SYSTEMS

SANDBOX

It does well in Assists, Blocks, Steals, TOs (assuming they wanted to use only integers), and 3s. It gives too much weight in rebounds, no weight in missed FGs and FTs, and uses a shooting bonus (which is extremely poorly designed, as a matter of fact). GRADE 5/10.

HOOPSCITY (FANTASY)

Does well in Rebounds, Assists (also assuming they used integers only), Blocks, TOs, 3s, and they avoid shooting bonuses. But it gives too low weight in steals (lower than TOs!) and no weight in missed FGs and FTs. GRADE 7/10.

HOOPSCITY (HEAD-TO-HEAD)

Same as Hoopscity Fantasy, but no 3s and they use bonuses. GRADE 5.5/10.

SMALL WORLD

The best system available in my opinion (analyzed in the previous section). Does very well in not avoiding half points. GRADE 8/10.

NBA.COM

Another very good system, a close 2nd. Under the luxury of the integer system they use, they fall short only in not including 3s made. GRADE 8/10.

June 8, 1999